where \(\kappa \stackrel{_{def}}{=}\int _{\mathbb {R}^{d}}e^{-\left\Vert \xi \right\Vert _{p}^{p}/p}d\xi =\frac{2^{d}\Gamma ^{d}(\frac{1}{p})}{p^{d-\frac{d}{p}}}\) 
and \(\xi \sim N_{p}(0,I_{d\times d})\)  (the \(p\) -generalized Gaussian
distribution). The rationale for taking into account the \(p\) -generalized
Gaussian smoothing \(U_{\mu }\)  rather than \(U\)  is that it typically
benefits from superior smoothness properties. In particular, \(U_{\mu }\) 
is smooth albeit \(U\)  is not. In addition, \(p\) -generalized Gaussian
smoothing is more generalized than Gaussian smoothing in the sense
that it contains all normal distribution when \(p=2\)  and all Laplace
distribution when \(p=1\) . This family of distributions allows for
tails that are either heavier or lighter than normal and in the limit
as well as containing all the continuous uniform distribution. More
importantly, we prove that a smoothing potential \(U_{\mu }(x)\)  is
actually smooth (gradient Lipschitz). This property is novel and potentially
useful in the optimization or sampling process, especially when the
potential exhibits some sort of weakly smooth behaviors. Here to simplify
the proof, we consider \(p\in \mathbb {R}^{+},\)  \(2\ge p\ge 1\)  and
some primary features of \(U_{\mu }\)  based on adapting those results
of [1]}.
